Finding the Stationary Positions of Point Vortices

One of the project’s objectives which are related to the experiment was to find if there exist stationary locations of the vortex – antivortex pair, both downstream and upstream of the disk. The stationary points can only be complex conjugates on a complex plane, in the upper, and in the lower half – plane. The stationary points are shown in the figure 3.1 below:

Figure 3.1: Point vortices around the disk. At the left are is the vortex – antivortex pair downstream and at the right is the vortex – antivortex upstream of the disk.

To find the stationary point positions the following cubic equation was used:

(3.1)

λ: is the non – dimensional circulation

This equation was solved with 3 ways. The first method was Cardano’s method using excel, also was solved using Matlab with root command and finally with fsolve command. The two m – files are shown in Appendix 1. The solution of this equation has given the values of and then these values where used in the following equation to find the values: (3.2) where

The range of λs that was used was from 0 to 5 with 0.1 intervals. So by the solution of the cubic equation the following graph was produced:

这是solved方程与3种。cardano’ method was the first method was also Excel的使用，solved using MATLAB命令与fsolve与根和最后的命令。两个are shown the M在附录1。the solution of this has given the values of方程，然后这些值在used in the values to find the following方程式（3.2）：在

λs that was used of the范围是从0到5 intervals和0.1。我知道by the solution of the following the三次方程为：图的制

图3.2坐标线：红色和蓝色线dashed固在上半平面作为函数λof the不维的循环

Analyse the motion of vortex – antivortex pairs of two vortices downstream of the cylinder

The next objective of the project was to see how the vortices move when particular stationary points chosen, the stationary points are depending on the non – dimensional circulation λ from the range that was mentioned before in section 3. To do this the Lagrangian Equations of Motion of Point Vortices in the Inviscid Flow around the Disk were used. The equations are shown below:

The equation 4.2 is for the upper half – plane vortex z1 and the equation 4.3 is for the lower half – plane vortex z2. Each of the vortices has two equations, one for real and one for imaginary solution, so the total equations are four. To analyse the motion, these four equations were solved with some initial conditions using Matlab.

First of all an m – file was made called Stream1down.m (Appendix 1) to set the ode equations to find how the vortices are move. Another m – file called Motion1down.m (Appendix 1) was made to set the required initial conditions to solve the ode equations. The stationary points in Motion1down.m are taken from the stationary1down.m (Appendix 1) and the λ value (q in m – files) that is required for Stream1down.m and in stationary1down.m is taken from the lambdainput.m (Appendix 1). One example of the graphs that were made is shown below, these three graphs are shown the trajectories for the vortex points of the vortex in the upper half-plane x1 vs y1 and the time – dependent trajectories x1(t) and y1(t) for λ = 1, 1.5, 1.8, 2 and 3.

From the time – depended figures 4.1.2 and 4.1.3, it is observed that for vortices initially located at the rear of the cylinder, the period of time during the vortex points remain close to their initial locations increases with the non – dimensional circulation λ. For example for λ = 1 the vortex point stays close to its initial location for approximately 5 non – dimensional units of time and for λ = 2 for approximately 35. When λ = 3 the vortex point remain close to its initial location for more than 70 no – dimensional units of time.